This is a follow up question to this one.

If $X$ is a metric space, denote by $C_u(X)$ the $C^\ast$-algebra of all bounded, uniformly continuous functions on $X$ (with the sup-norm).

Do we have $C_u(X_1 \times X_2) = C_u(X_1) \hat{\otimes} C_u(X_2)$?

(Maybe it is more natural to use uniform spaces instead of metric spaces, but I think that the answer does not depend on this.)